Since infinite homogeneous trees are discrete analogues of the hyperbolic disc, these trees are a natural environment for studying free group actions and also spectra of transition operators. We outline their introduction in harmonic analysis, discrete potential theory and random walks, and review old and new results on strictly related subjects: the spectrum of the Laplace operator on a homogeneous tree, uniformly bounded representations of free groups, boundary behaviour of harmonic functions, nearest neighbour and finite step transition operators, and the Poisson and Martin boundaries.
A great amount of deep progress on representations of free groups and on spectra of transition operators on groups and graphs has followed these preliminary steps in the course of the years: substantial results have been obtained by K. Aomoto, T. Steger, S. Lalley, W. Woess, V. Kajmanovich, L. Saloff-Coste, Th. Coulhon, N.Th. Varopoulos, T. Nagnibeda and many others, not to mention the deep theory of Gromov and hyperbolic graphs, but all these results are beyond the scope of this presentation.